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------------------------------------------------ Subtract and simplify the expression:

[tex]
\[
\left(3x^4 + 9x^3 - 5\right) - \left(4x^2 - 6x + 1\right)
\]
[/tex]

[tex]
\[
= 3x^4 + 9x^3 - 5 - 4x^2 + 6x - 1
\]
[/tex]

Simplified answer:

[tex]
\[
3x^4 + 9x^3 - 4x^2 + 6x - 6
\]
[/tex]

To subtract the two polynomials [tex]\((3x^4 + 9x^3 - 5)\)[/tex] and [tex]\((4x^2 - 6x + 1)\)[/tex], we will follow a straightforward step-by-step process:

1. Write Down the Polynomials:
- The first polynomial is [tex]\(3x^4 + 9x^3 - 5\)[/tex].
- The second polynomial is [tex]\(4x^2 - 6x + 1\)[/tex].

2. Set Up the Subtraction:
- We need to subtract the second polynomial from the first one. This can be written as:
[tex]\[
(3x^4 + 9x^3 - 5) - (4x^2 - 6x + 1)
\][/tex]

3. Distribute the Negative Sign:
- Distribute the negative sign across the second polynomial:
[tex]\[
3x^4 + 9x^3 - 5 - 4x^2 + 6x - 1
\][/tex]

4. Combine Like Terms:
- Group and combine the like terms:
- The term with [tex]\(x^4\)[/tex] is [tex]\(3x^4\)[/tex].
- The term with [tex]\(x^3\)[/tex] is [tex]\(9x^3\)[/tex].
- The term with [tex]\(x^2\)[/tex] is [tex]\(-4x^2\)[/tex].
- The term with [tex]\(x\)[/tex] is [tex]\(6x\)[/tex].
- The constant terms are [tex]\(-5\)[/tex] and [tex]\(-1\)[/tex], which combine to [tex]\(-6\)[/tex].

5. Write the Simplified Polynomial:
- After combining, the simplified result of the subtraction is:
[tex]\[
3x^4 + 9x^3 - 4x^2 + 6x - 6
\][/tex]

This polynomial [tex]\(3x^4 + 9x^3 - 4x^2 + 6x - 6\)[/tex] is your final answer, representing the subtraction of the given polynomials.